Answers and Explanations 251
Answers
301–400
Next, apply the quotient rule, making sure to use the chain rule when taking the
derivative of the denominator:
f x
x
x
x
x
x
' ( ) =
−
(
) ( )−
−
(
) −
( )
(
)
−
(
)
(
)
=
−
(
)
−
−
3 2
1
1
2
3 2
2
3 2
3 2
3
1 2
1 2
1 2
2
1 2
− −
(
)+
(
)
−
(
)
=
−
−
(
)
2
3 2
3
3 2
3 2
x x
x
x
x
362.
4 sec
2 
x tan x
First rewrite the function:
f x
x
x
x
x
( ) sec
tan
sec
t an
=
+
= (
) + (
)
2
2
2
2
Applying the chain rule gives you the derivative:
=
+
=
x
x
x
x
x
x
x
( )
(sec )sec tan
(tan )sec
sec tan
2
2
4
2
2
x
f '
363.
1
1
2
3 2
+
(
)
x
Recall that the chain rule states
d
dx
f g x
f g x g x
( )
( ) ( )
(
)= (
)
'
'
and that the quotient rule states
d
dx
f x
g x
g x f x f x g x
g x
( )
( )
( ) ( ) ( ) ( )
( )





 =
−
[
]
2
'
'
Rewrite the function using exponential notation:
f x
x
x
x
x
( ) =
+
=
+
(
)
1
1
2
2
1 2
Apply the quotient rule and the chain rule to get the derivative:
=
+
(
)
−
+
(
)
(
)
+
(
)
(
)
=
+
(
)
−
−
x
x
x
x
x
x
( )
( )
( )
1
1
1
2
1
2
1
1
1
2
1 2
2
1 2
2
1 2 2
2
1 2
+ +
(
) −
(
)
+
(
)
=
+
(
)
x
x
x
x
2
2
2
2
3 2
1
1
1
x
f '